Optimal. Leaf size=122 \[ \frac{1}{7} d x^7 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{5} c x^5 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 x+\frac{1}{3} a c^2 x^3 (3 a d+2 b c)+\frac{1}{9} b d^2 x^9 (2 a d+3 b c)+\frac{1}{11} b^2 d^3 x^{11} \]
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Rubi [A] time = 0.0538846, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {373} \[ \frac{1}{7} d x^7 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{5} c x^5 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 x+\frac{1}{3} a c^2 x^3 (3 a d+2 b c)+\frac{1}{9} b d^2 x^9 (2 a d+3 b c)+\frac{1}{11} b^2 d^3 x^{11} \]
Antiderivative was successfully verified.
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Rule 373
Rubi steps
\begin{align*} \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx &=\int \left (a^2 c^3+a c^2 (2 b c+3 a d) x^2+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^4+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^6+b d^2 (3 b c+2 a d) x^8+b^2 d^3 x^{10}\right ) \, dx\\ &=a^2 c^3 x+\frac{1}{3} a c^2 (2 b c+3 a d) x^3+\frac{1}{5} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^5+\frac{1}{7} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^7+\frac{1}{9} b d^2 (3 b c+2 a d) x^9+\frac{1}{11} b^2 d^3 x^{11}\\ \end{align*}
Mathematica [A] time = 0.0209226, size = 122, normalized size = 1. \[ \frac{1}{7} d x^7 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{5} c x^5 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 x+\frac{1}{3} a c^2 x^3 (3 a d+2 b c)+\frac{1}{9} b d^2 x^9 (2 a d+3 b c)+\frac{1}{11} b^2 d^3 x^{11} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 125, normalized size = 1. \begin{align*}{\frac{{b}^{2}{d}^{3}{x}^{11}}{11}}+{\frac{ \left ( 2\,ab{d}^{3}+3\,{b}^{2}c{d}^{2} \right ){x}^{9}}{9}}+{\frac{ \left ({a}^{2}{d}^{3}+6\,abc{d}^{2}+3\,{b}^{2}{c}^{2}d \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{a}^{2}c{d}^{2}+6\,ab{c}^{2}d+{b}^{2}{c}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,{a}^{2}{c}^{2}d+2\,ab{c}^{3} \right ){x}^{3}}{3}}+{a}^{2}{c}^{3}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.989701, size = 167, normalized size = 1.37 \begin{align*} \frac{1}{11} \, b^{2} d^{3} x^{11} + \frac{1}{9} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{9} + \frac{1}{7} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{7} + a^{2} c^{3} x + \frac{1}{5} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{5} + \frac{1}{3} \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.08426, size = 296, normalized size = 2.43 \begin{align*} \frac{1}{11} x^{11} d^{3} b^{2} + \frac{1}{3} x^{9} d^{2} c b^{2} + \frac{2}{9} x^{9} d^{3} b a + \frac{3}{7} x^{7} d c^{2} b^{2} + \frac{6}{7} x^{7} d^{2} c b a + \frac{1}{7} x^{7} d^{3} a^{2} + \frac{1}{5} x^{5} c^{3} b^{2} + \frac{6}{5} x^{5} d c^{2} b a + \frac{3}{5} x^{5} d^{2} c a^{2} + \frac{2}{3} x^{3} c^{3} b a + x^{3} d c^{2} a^{2} + x c^{3} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.080935, size = 136, normalized size = 1.11 \begin{align*} a^{2} c^{3} x + \frac{b^{2} d^{3} x^{11}}{11} + x^{9} \left (\frac{2 a b d^{3}}{9} + \frac{b^{2} c d^{2}}{3}\right ) + x^{7} \left (\frac{a^{2} d^{3}}{7} + \frac{6 a b c d^{2}}{7} + \frac{3 b^{2} c^{2} d}{7}\right ) + x^{5} \left (\frac{3 a^{2} c d^{2}}{5} + \frac{6 a b c^{2} d}{5} + \frac{b^{2} c^{3}}{5}\right ) + x^{3} \left (a^{2} c^{2} d + \frac{2 a b c^{3}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12057, size = 177, normalized size = 1.45 \begin{align*} \frac{1}{11} \, b^{2} d^{3} x^{11} + \frac{1}{3} \, b^{2} c d^{2} x^{9} + \frac{2}{9} \, a b d^{3} x^{9} + \frac{3}{7} \, b^{2} c^{2} d x^{7} + \frac{6}{7} \, a b c d^{2} x^{7} + \frac{1}{7} \, a^{2} d^{3} x^{7} + \frac{1}{5} \, b^{2} c^{3} x^{5} + \frac{6}{5} \, a b c^{2} d x^{5} + \frac{3}{5} \, a^{2} c d^{2} x^{5} + \frac{2}{3} \, a b c^{3} x^{3} + a^{2} c^{2} d x^{3} + a^{2} c^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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